Lemma 3.2

Let $L$ be a complete lattice and let $a\mapsto a^\perp$ be a function $L\to L$ satisfying, for all $a,b\in L$, $a\wedge a^\perp=\bot$, $a\vee a^\perp=\top$, $(a\wedge b)^\perp=a^\perp\vee b^\perp$, $(a\vee b)^\perp=a^\perp\wedge b^\perp$, and $(a^\perp)^\perp=a$, and additionally satisfying $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$ whenever $b\le a$ and $c\le a^\perp$. Define a quasipoint of $L$ to be an inclusion-maximal nonempty subset $B\subseteq L$ that excludes $\bot$ and such that any two members have a common lower bound belonging to $B$. Let $\mathcal Q(L)$ be the set of quasipoints, equipped with the topology generated by $Q_a=\{B\in\mathcal Q(L):a\in B\}$. Then the following are equivalent: for every nonempty upward-directed subset $S\subseteq L$, $\overline{\bigcup_{a\in S}Q_a}=Q_{\sup S}$; and for every $a\in L$ and every nonempty upward-directed subset $S\subseteq L$, $a\wedge\sup S=\sup\{a\wedge b:b\in S\}$. Here upward-directed means that any two members of $S$ have a common upper bound in $S$; neither condition imposes a requirement on the empty subset.